...In 2015, Mr. Kelley tried unsuccessfully to get a Texas license to carry a handgun — a failure that was tied to an animal cruelty charge against him the year before. While he was living in Colorado Springs, he was charged with a misdemeanor for beating his dog. A judge imposed a deferred judgment, and the charge was dismissed two years later, after Mr. Kelley completed a court-ordered period of probation.

When Mr. Kelley applied for a license to carry, a check with the national criminal background databases did not turn up his military conviction. But it did reveal an entry from El Paso County, Colo., the Texas Department of Public Safety said on Wednesday — an apparent reference to the animal cruelty case.

The department said it notified him that his application was delayed because of a “possibly disqualifying issue,” and asked him for more information regarding his application. He did not respond, so the application was denied....